Viscous Fingering In Complex Magnetic Fluids: Weakly Nonlinear Analysis, Stationary Solutions And Phase-field Models

PhD THESIS

VISCOUS FINGERING IN COMPLEX

MAGNETIC FLUIDS: WEAKLY

NONLINEAR ANALYSIS, STATIONARY

SOLUTIONS AND PHASE-FIELD

MODELS

Sérgio Henrique Albuquerque Lira

Recife - PE, Brazil

2014

Acknowledgements

First of all, I thank prof. José Miranda for all the advising support and confidence in our research. It has been a pleasure to work with him during all these years, to be part of his group in Recife and also to be his personal friend.

I also thank prof. Jaume Casademunt for receiving me so well at the Universitat de Barcelona. I am very grateful for being able to work there on the lamellar fragment problem under his su-pervision during my one year sandwich stage. Living in Barcelona was indeed an amazing experience and made me grow personally and academically.

I thank Rafael Oliveira for all the collaboration in obtaining the exact stationary solutions presented in Chapter 3, and also for further numerical discussions. I thank João Fontana for the collaboration on the yield stress problem of Chapter 4. I thank Carles Blanch-Mercader very much for all the support on numerical and analytical discussions that led us to the results presented in Chapter 5.

I am also thankful to Hermes Gadêlha for all the nice discussions about biological problems. To my colleagues Eduardo and Chico for their joyful company and fruitful discussions. To the Fractal members Victor, Tiago, Leo and Hugo, for organizing great meetings. To everyone that helped me at the UB and made my staying there be so pleasant, especially to my office colleagues Xumeu, Elisenda and Albert.

In addition, I would like thank all the members of the jury for accepting being part of this process and useful comments.

Finally, my special thanks to my wife Fernanda for unconditional love and support. For financial support I thank to CNPq and INCT-FCx.

It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment.

Resumo

Nesta Tese são empregadas técnicas analíticas e numéricas para investigar o fenômeno de for-mação de dedos viscosos entre fluidos imiscíveis confinados quando um destes fluidos é um fluido magnético complexo. Diferentes tipos de esquemas geométricos efetivamente bidimen-sionais foram investigados. Duas situações distintas são tomadas com relação à natureza da amostra de fluido magnético: um fluido newtoniano usual, e um fluido magneto-reológico que apresenta um yield stress dependente da intensidade do campo magnético. Equações gover-nantes adequadas são derivadas para cada um dos casos. Para obter um entendimento analítico dos estágios iniciais da evolução temporal da interface foi empregada uma análise fracamente não-linear de modos acoplados. Este tipo de análise acessa a estabilidade de uma interface inicialmente perturbada e também revela a morfologia dos dedos emergentes. Em algumas circunstâncias soluções estacionárias podem ser encontradas mesmo na ordem não-linear mais baixa. Nesta situação é feita uma comparação de algumas destas soluções com soluções es-táticas totalmente não-lineares obtidas através de um formalismo de vortex-sheet na condição de equilíbrio. Em seguida foi desenvolvido um modelo de phase-field aplicado a fluidos mag-néticos que é capaz de simular numericamente a dinâmica totalmente não-linear do sistema. O modelo consiste em introduzir uma função auxiliar que reproduz uma interface difusa de espessura finita. Utilizando esta ferramenta também é possível estudar um complexo problema de dedos viscosos de origem biológica: o fluxo de actina como um fluido ativo dentro de um fragmento lamelar.

Palavras-chave: Formação de dedos viscosos. Ferrofluido. Fluido magneto-reológico. Yield

Abstract

In this thesis, analytical and numerical approaches are employed in order to investigate the phenomenon of viscous fingering between confined immiscible fluids when one of the fluids is a complex magnetic fluid. Different types of effectively two-dimensional geometrical setups and applied magnetic field configurations are investigated. Two distinct situations are taken for the nature of magnetic fluid sample: a regular Newtonian ferrofluid, and a magnetorheological fluid that presents a magnetic field-dependent yield stress. Suitable governing equations are derived for each one of the cases. To obtain analytical insight about early stages of the time evolving interface we employ a weakly nonlinear mode-coupling approach. This kind of anal-ysis accesses the stability of an initially perturbed interface, and also reveals the morphology of the emerging fingers. At some circumstances, stationary solutions may be found already at lowest nonlinear order. In this context, we compare some of these solutions to fully nonlinear steady profiles obtained by using a vortex-sheet formalism at the equilibrium condition. More-over, we develop a phase-field model applied to magnetic fluids that is capable of numerically simulate the fully nonlinear dynamics of the system. The model consists on introducing an auxiliary function that reproduces a diffuse interface of finite thickness. By utilizing this tool we are also able to study a complex viscous fingering problem of biological origin: the flow of actin as an active fluid inside of a lamellar fragment.

Keywords: Viscous fingering. Ferrofluid. Magnetorheological fluid. Yield stress. Phase-field.

List of Figures

1.1 Schematic top views of the Saffman-Taylor instability based on experiments at

the channel geometry [3]. 15

1.2 Ferrofluid viscous fingering patterns produced by different magnetic field con-figurations. The left panel [19] depicts the labyrinthine instability formed when a uniform magnetic field is applied perpendicularly to the plates. The mid and right panels [20] show spiral and protozoan-like shapes that arise when a

rotat-ing magnetic field is added to the perpendicular field. 17

1.3 Magnetorheological fluid sample at the abscence of an applied magnetic field

(left) and at the presence of a magnet (right). 18

1.4 Figure extracted from [35]. The left panel shows a fairly circular actin lamellar fragment that does not propagate. When perturbed this fragment may acquire the steady shape in the right panel that propagates upwards, where the white

bar in the bottom gives a one micrometer scale. 19

2.1 Schematic illustration of the vertical Hele-Shaw cell setup. Fluid 1 is a fer-rofluid (shaded region), while fluid 2 is nonmagnetic. The densities and vis-cosities of the fluids are respectively denoted byρj, andηj, where j= 1 and 2.

A uniform magnetic field H0is applied along the positive y direction, and the

acceleration of gravity points downward (g_{= −gˆy). The cell has thickness b,} and interfacial perturbations are represented byζ =ζ(x,t). 25 2.2 Linear growth rate λ(k) as a function of the wave number k for NG= 1.44,

and three different values of the magnetic Bond number NB. The critical (kc),

fastest growing (k∗), and threshold (kt) wave numbers are also indicated. The

2.3 Time evolution of the interface shape. The shaded region represents the fer-rofluid pattern morphology at time t = 2. The interface profile for t > 2 is

indistinguishable from the one shown at t= 2. 33

2.4 Time evolution of the perturbation amplitudes ak(t) and a2k(t) for the evolving

interface depicted in Fig. 2.3. It is clear that both amplitudes eventually tend to

stationary values. 34

2.5 Schematic configuration of the parallel flow in a vertical Hele-Shaw cell. The lower fluid is a ferrofluid, while the upper fluid is nonmagnetic. An exter-nal uniform magnetic field H0 is applied making an angleα with the initially

undisturbed interface separating the fluids. 38

2.6 The real part of the linear growth rateλ(k) as a function of the wave number k for NG= 1.4. Continuous (dashed) curves refer to NB = 20 (NB = 30). For

each NB we plot curves for three values of the angle α, where lighter gray

curves correspond to higher values ofα. 43

2.7 Dominant wave number kmax as a function of the angle α for NG= 1.4 and

three different values of NB. The dots indicate the critical values ofα below

which the interface is stable. 44

2.8 Numerical time evolution of the absolute value of the perturbation amplitudes for the fundamental mode (2pζ_kζ_−k) and its first harmonic (2pζ_2kζ_−2k). The parameters considered correspond to the continuous dark gray curve in Fig. 2.6 (NG= 1.4, c0= 0.5, NB = 20 andα = 1.30). As time grows, the amplitudes

tend to saturate and reach stationary values indicating the propagation of an

unchanged shape profile. 46

2.9 Propagating wave profile for c0= 0.5, NB= 20 andα=π/2, resulting in vf =

0.56. The profile and velocity are reflected in relation to the y axis if we perform

2.10 (a) Propagating wave profile for NB = 20 and α = 1.17, resulting in vf =

17.83.(b) Propagating wave profile for NB = 20 and α = 1.30, resulting in

vf = 16.98. The profile and velocity are reflected in relation to the y axis if

we perform the transformationα′=π₋α. Note that here c0= 0. 48

2.11 Propagating final velocity vf as a function ofα for c0= 0, NB = 20 (black),

NB= 25 (dark gray) and NB = 30 (light gray). The dotted curves depict the

linear prediction of the fastest growing mode phase velocity, the solid curves correspond to the analytical weakly nonlinear prediction, and the dots show the velocities obtained by numerically evaluating the time evolution of Eqs. (2.31) and (2.32). The dashed vertical lines indicate the critical values ofα. 51 3.1 Schematic illustration of a Hele-Shaw cell of thickness b containing an initially

circular droplet (dashed curve) of a MR fluid, surrounded by a nonmagnetic fluid. The anti-Helmholtz coils produce a magnetic field H pointing radially outwards in the plane of the cell. Fingering interfacial patterns arise due to the

action of the radial magnetic field. 56

3.2 Behavior of the finger tip function T(2n, n) as the magnetic Bond number NB

is varied, for R= 0.9,χ= 0.5, and two different values of the zero field yield stress parameter: S0= 57.6 (solid curves), and S0= 128 (dashed curves). For

each value of S0, three increasing magnitudes for the magnetic field-dependent

yield stress parameter are used: S= 60 (black), S = 67 (dark gray), and S = 74

(light gray). 65

3.3 Typical stationary shape solutions for NB = 256, χ = 0.5, ψ0 =π/2, r0= 1, S0= 57.6, and (a) S = 105.13, (b) S = 100.13, (c) S = 93.13, and (d) S = 84.13. 69

3.4 Typical stationary shape solutions for NB = 256, χ = 0.5, ψ0 =π/2, r0= 1, S0= 128, and (a) S = 81.67, (b) S = 76.67, (c) S = 69.67, and (d) S = 60.67. 71

3.5 Gallery of possible patterns for increasingly larger values of the magnetic Bond number NB. It is assumed that χ = 0.5, ψ0=π/2, r0= 1, a = −6.36, S0=

38.02, S = 23.80, and (a) NB= 109.85, (b) NB= 126.57, (c) NB= 148.06, and

3.6 Sketch of a rotating Hele-Shaw cell of thickness b containing an initially circu-lar magnetic fluid droplet of radius R. The in-plane azimuthal magnetic field H is produced by a long wire carrying an electric current I. The cell rotates with constant angular velocityΩaround an axis coincident with the wire. 74 3.7 Typical stationary shape solutions for a Newtonian ferrofluid droplet (S0= S =

0), and three different values of the rotational Bond number N_Ω. The intensity of the magnetic Bond number NB increases from left to right. 81

3.8 Typical stationary shape solutions for a MR fluid droplet, and two different values of the zero field yield stress parameter S0. Both NB and NΩ are kept

fixed, while the field-induced yield stress parameter S increases from left to right. 83 3.9 Newtonian ferrofluid situation. Left panel: cosine Fourier amplitudes as a

func-tion of the azimuthal mode number n. Right panel: comparison between exact and weakly nonlinear (WNL) solutions for the steady interface shape. 85 3.10 Magnetorheological fluid situation. Left panel: cosine Fourier amplitudes as a

function of the azimuthal mode number n. Right panel: comparison between exact and weakly nonlinear (WNL) solutions for the steady interface shape. 86 3.11 Left panel: 2D phase portrait. Right panel: time evolving weakly nonlinear

patterns (solid interfaces) for two different initial conditions (1 and 4). The dashed interface represents the saddle point associated to the MR fluid weakly

nonlinear pattern shown in Fig. 3.10. 87

4.1 Schematic configuration of radial flow in a Hele-Shaw cell. The inner fluid is Newtonian and has negligible viscosity. The outer fluid is a yield stress fluid. The unperturbed fluid-fluid interface (dashed curve) is a circle of radius R. All

5.1 Phase-field simulation of an initially perturbed ferrofluid droplet subjected to a radial magnetic field. Upper panels: phase-field plots in color scale for three different times, whereθ = +1 (θ _{= −1) corresponds to the inner (outer) fluid} phase. Lower panels: stream function plots for the correspondent times in the

upper plots. 103

5.2 Phase-field simulation of an initially perturbed cell fragment. Upper panels: phase-field plots in color scale for three different times, where θ = +1 (θ = −1) corresponds to the inner (outer) fluid phase. Mid panels: stream function plots for the correspondent times in the upper plots. Lower panels: auxiliaryφ function plots for the correspondent times in the upper plots. Boundary

condi-tions are set asθ _{= −1.0,}ψ = 0.0 andφ = 0.0. 112

5.3 Phase-field simulation of an initially perturbed cell fragment. Upper panels: phase-field plots in color scale for three different times, where θ = +1 (θ = −1) corresponds to the inner (outer) fluid phase. Mid panels: stream function plots for the correspondent times in the upper plots. Lower panels: auxiliaryφ function plots for the correspondent times in the upper plots. Boundary

Contents

1 Introduction 15

1.1 Pattern formation in viscous fingering 15

1.2 Darcy’s law for Newtonian fluids 19

1.3 Thesis outline 21

2 Field-induced patterns in ferrofluids 23

2.1 Normal-field instability in confined ferrofluids 24

2.1.1 Mode coupling strategy 25

2.1.2 Pattern morphology and nonlinear stability of stationary patterns 29

2.2 Nonlinear traveling waves in confined ferrofluids 35

2.2.1 Governing equations and analytical calculations 37

2.2.2 Linear regime 43

2.2.3 Weakly nonlinear dynamics 45

2.2.4 Analytical approach to steady solutions 49

2.2.5 Nonlinear velocity calculation 52

3 Field-induced patterns in MR fluids 55

3.1 Radial magnetic field 56

3.1.1 Physical problem and governing equations 57

3.1.2 Linear stability and weakly nonlinear dynamics 62

3.1.3 Exact stationary solutions 67

3.2 Stationary shapes of confined rotating magnetic liquid droplets 72

3.2.1 Specification of the moving boundary problem 76

3.2.2.1 Access to fully nonlinear steady shapes 79

3.2.2.2 Newtonian ferrofluid 81

3.2.2.3 Magnetorheological fluid 83

3.2.3 Mode-coupling approach and the stability of the exact solutions 84

4 Darcy’s law formulation for Bingham fluids 91

5 Phase-field approach 97

5.1 Viscous fingering in usual Newtonian fluids 98

5.2 Viscous fingering in magnetic fluids 101

5.2.1 Ferrofluid in a radial magnetic field 102

5.3 Actin-based motility of lamellar fragments 105

5.3.1 Problem formulation: sharp interface equations 106

5.3.2 Phase-field equations 108

5.3.2.1 Dimensionless parameters 110

5.3.2.2 Numerical implementation 110

6 Conclusions 116

6.1 Conclusions from Chapter 2 116

6.2 Conclusions from Chapter 3 117

6.3 Conclusions from Chapter 4 119

6.4 Conclusions from Chapter 5 119

C

1

Introduction

1.1

Pattern formation in viscous fingering

Classical viscous fingering pattern formation takes place when a less viscous fluid displaces a more viscous one in the confined geometry of two narrowly spaced parallel plates, the so called Hele-Shaw (H-S) cell [1]. The initially flat interface between these two immiscible fluids becomes unstable, so broad perturbations arise and tend to grow, what is commonly referred to as the Saffman-Taylor instability (see Fig. 1.1). This problem has produced a lot interest for physicists and engineers during several decades due to its prototypical character with many theoretical and practical implications. Mathematically, it is defined as a nonlocal moving boundary problem of Laplacian growth, and it is intimately related to a variety of groundbreaking phenomena, such as dendritic growth, oil recovery, etc [2]. In practice, it is an excellent laboratory because of its relative simplicity both experimentally and in its theoretical formulation.

Figure 1.1: Schematic top views of the Saffman-Taylor instability based on experiments at the channel geometry [3].

Despite of being most known by its classical viscosity-driven setup, the Saffman-Taylor instability may also be driven by other kinds of mechanisms. For instance, the rotating

Hele-Shaw problem is a variation of the traditional instability [1, 2], in which the cell rotates, and the competition between centrifugal and capillary forces results in interface destabilization. During the last two decades different aspects of the problem have been investigated, including the development of zero surface tension time-dependent exact solutions [4, 5, 6], the consideration of miscible fluid displacements [7], the dependence of pattern morphologies on viscous [8, 9] and wetting [10] effects, the influence of Coriolis force on the interfacial dynamics [11, 12], and the occurrence of complex pinch-off phenomena [13].

Another suggestive variant of the Hele-Shaw problem with usual viscous fluids considers that at least one of the fluids is a ferrofluid [14, 15], a superparamagnetic liquid which promptly responds to even modest magnetic stimuli. This property turns out to be very interesting since it introduces the possibility of generating pretty different viscous fingering patterns by adjust-ing an applied magnetic field. One compelladjust-ing example of pattern-formadjust-ing systems in confined ferrofluids is related to the labyrinthine instability [16, 17, 18], in which highly branched struc-tures are formed when a magnetic field is applied perpendicularly to the plates of a Hele-Shaw cell (see left panel in Fig. 1.2). Beautiful spiral patterns and amazing protozoan-like shapes can also arise when a rotating magnetic field is added to the perpendicular field setup [20] (see right panel in Fig. 1.2). The emergence of peculiar diamond-ring-shaped structures has been detected in centrifugally-driven Hele-Shaw flows under the action of an azimuthal magnetic field [21]. In addition, quite regular n-fold symmetric shapes emerge in both immiscible [22] and miscible [23] ferrofluids when perpendicular and azimuthal magnetic fields are applied simultaneously. Finally, the development of starfish-like morphologies has been recently pre-dicted if a radial magnetic field configuration is used [24].

In contrast to what happens to ferrofluids, the investigation of Hele-Shaw pattern formation with magnetorheological (MR) fluids has been amply overlooked. Magnetorheological flu-ids consist of much larger, micronsized magnetized particles dispersed in aqueous or organic carrier liquids. The unique feature of this kind of magnetic fluid is the abrupt change in its viscoelastic properties upon the application of an external magnetic field [25, 26, 27, 28, 29] (see Fig. 1.3). In the absence of an applied field ("off" state) the magnetized particles in the sus-pension are randomly distributed, so that MR fluids appear similar to usual nonmagnetic fluids.

Figure 1.2: Ferrofluid viscous fingering patterns produced by different magnetic field configu-rations. The left panel [19] depicts the labyrinthine instability formed when a uniform magnetic field is applied perpendicularly to the plates. The mid and right panels [20] show spiral and protozoan-like shapes that arise when a rotating magnetic field is added to the perpendicular field.

However, when a magnetic field is applied ("on" state) the large particles suspended in the fluid interact, and tend to align and link together along the field’s direction, creating long particle chains, columns, and other more complex structures. Interestingly, the formation of such struc-tures restrict the motion of the fluid, allowing it to display a solidlike behavior. A MR fluid can be characterized by its yield stress, which measures the strength of the field-induced structures formed.

Despite all the efforts and important results obtained by researchers on the development of viscous fingering in Newtonian Hele-Shaw flows (i. e., constant viscosity fluid flow), the pattern forming dynamics with yield stress fluids, even at the nonmagnetic case, has been rel-atively underlooked. In contrast to Newtonian fluids, yield stress fluids [30, 31] can support shear stresses without flowing. As long as the stress remains below to a certain critical value they do not flow, but respond elastically to deformation. So, such materials possess properties of both viscous fluids and elastic solids, behaving like a “semi-solid". On the theoretical side, a linear stability analysis of the Saffman-Taylor problem in rectangular and radial cells with yield stress fluids [32] has predicted that the instability can be drastically modified. On the ex-perimental arena some interesting findings have been disclosed in channel geometry [33, 34]: depending on whether viscous effects or yield stresses dominates, fractal patterns, or ramified

Figure 1.3: Magnetorheological fluid sample at the abscence of an applied magnetic field (left) and at the presence of a magnet (right).

structures where multiple fingers propagate in parallel may arise.

Another remarkable scenario in which viscous fingering takes place is that of biological fluids. Recently, it has been shown that in appropriate circumstances the flow of actin in lamel-lar fragments satisfies Darcy’s law in an effectively two-dimensional geometry [35, 36], thus reducing the dynamics to a free-boundary problem similar to that of viscous-fingering in Hele-Shaw cells, but with different boundary conditions [37]. The comprehension of this mechanism is of major importance for one to attain shape polarization that allows cell motility and thread-milling. As shown in Fig. 1.4, a lamellar cell fragment may undergo on a shape transition that produces threadmilling by actin polymerization.

In order to fill some of the gaps exposed above, this Thesis proposes a theoretical study about the viscous fingering phenomenon in complex magnetic fluids. And by complex mag-netic fluids we comprehend ferrofluids, MR fluids, yield stress fluids and lamellar fragments. During its development, we make use of analytical and numerical tools to elucidate the main aspects of the dynamics and morphology of such interfacial pattern formations.

Figure 1.4: Figure extracted from [35]. The left panel shows a fairly circular actin lamellar fragment that does not propagate. When perturbed this fragment may acquire the steady shape in the right panel that propagates upwards, where the white bar in the bottom gives a one micrometer scale.

1.2

Darcy’s law for Newtonian fluids

We now make a brief derivation of the governing equations for classical viscous fingering in regular Newtonian fluids. By understanting this simpler situation we will be able to describe more complex systems in our further investigation.

Consider two immiscible and incompressible fluids of viscosity ηj and densityρj (where

j= 1, 2 labels the different fluids) are placed between two parallel rigid plates of transversal separation b. The equation that governs the hydrodynamic flow of such fluids is the Navier-Stokes equation ρj ∂ uj ∂t + (uj·∇)uj  = −∇Pj+ηj∇2uj, (1.1)

where u is the 3-D fluid velocity and P is its 3-D pressure field. Eventual extra forces would appear added at the right hand side of Eq. (1.1). Since we are dealing with an effective two-dimensional problem, we may reduce the 3-D flow to an equivalent 2-D one by gap-averaging Eq. (1.1) at the direction perpendicular to the plates. This is done by considering non-slip boundary conditions at the plates and taking, thus, the velocity profile as being parabolic at

the transversal direction. We also assume that the lubrication approximation is valid, i. e., that inertial terms in the right hand side of Eq. 1.1 are negligible when compared to viscous contributions. By following these steps we get Darcy’s law

vj= −

b2

12ηj

∇pj, (1.2)

where v, p and ∇are now the gap-averaged velocity, pressure and 2-D gradient operator, re-spectively. This is also the equation that describes fluid flow in porous media.

To complete the description of the moving boundary problem between two fluids we must also take into account the boundary conditions across the fluid-fluid interface

(v2− v1) · n = 0, (1.3)

(v2− v1) · s =

2b2 12(η1+η2)

s_{· [}∇(p1− p2) + A(v2+ v1)], (1.4)

where A= (η1−η2)/(η1+η2) is the viscous contrast. Eq. (1.3) is the kinematic boundary

condition and it imposes that both fluids have the same normal velocity component at the in-terface, since the normal interface velocity itself is given by Vn= v2· n = v1· n. On the other

hand, Eq. (1.4) says that the tangential velocity components are discontinuous at the interface, and we say that (v2− v1) · s is the interface vortex-sheet. This is the only source of vorticity

in the problem, since at each bulk phase Eq. (1.2) guarantees that the flow is vortex-free. The remaining ingredient is the pressure jump condition

p1− p2=σκ, (1.5)

whereσ is the surface tension between the fluids andκ is the interface in-plane curvature. To conclude we point out that since the fluids are incompressible

∇_{· v}j= 0, (1.6)

problem is completely determined in terms of the velocity field by Eqs. (1.6), (1.3) and (1.4).

1.3

Thesis outline

In Chapter 2 we investigate the problem of a ferrofluid confined in a vertical Hele-Shaw cell and subjected to an in-plane uniform magnetic field. In Section 2.1 we take the particular case where the applied magnetic field is normal to the initially flat interface and show the main results of Ref. [38]. In Section 2.2 we subject both upper and lower fluids to a parallel flow and let the applied magnetic field to make a tilting angle with the initial interface, as in Ref. [39]. In both cases, we perform a weakly nonlinear analysis that is able to reproduce the morphology of such pattern formation phenomenon at lowest nonlinear order. A mode-coupling theory is used to compare the early nonlinear evolution of the interface with asymptotic shapes obtained when relevant forces equilibrate. Our nonlinear results indicate that the time-evolving shapes tend to approach stable stationary solutions for the normal magnetic field case, and propagating steady nonlinear waves for the tilted field.

In Chapter 3 we study the behavior of a magnetorheological fluid droplet confined to a Hele-Shaw cell in the presence of an applied magnetic field. In Section 3.1 we consider the case of an in-plane radially increasing external magnetic field case, as explored in Ref. [40]. In Section 3.2 we take a rotating H-S cell under the presence of an azimuthal field produced by a current carrying wire, as in Ref. [41]. Interfacial pattern formation is investigated by consider-ing the competition among capillary, viscoelastic, and magnetic forces. The contribution of a magnetic field-dependent yield stress is taken into account. Linear stability analysis reveals the stabilizing role played by yield stress. On the other hand, a mode-coupling approach predicts that the resulting fingering structures should become less and less sharp as yield stress effects are increased. By employing a vortex-sheet formalism we have been able to identify a family of exact stationary solutions of the problem. A weakly nonlinear approach is employed to ex-amine this fact and to gain analytical insight into relevant aspects related to the stability of such exact stationary solutions.

In Chapter 4 we report analytical results contained in Ref. [42] for the development of inter-facial instabilities in a radial Hele-Shaw cell in which a nonmagnetic yield stress fluid is pushed by a Newtonian fluid of negligible viscosity. By dealing with a gap averaging of the Navier-Stokes equation, we derive a Darcy-law-like equation for the problem, valid in the regime of high viscosity compared to yield stress effects, and that accounts for a general yielding direc-tion.

In Chapter 5 we present phase-field numerical models inspired by Ref. [43] that simulate viscous fingering in a H-S cell. In Section 5.2 we develop such diffuse interface method for the case of magnetic fluids, and in Section 5.3 this is done for the case of lamellar fragments.

C

2

Field-induced patterns in ferrofluids

A ferrofluid is a stable colloidal suspension of nanometric magnetic particles dispersed in a nonmagnetic liquid carrier [14, 15] which responds paramagnetically to applied magnetic fields. The most remarkable feature of this material is the fact that it combines the fluidity of liquids and the magnetic properties typical of solids. The magnetic susceptibility of ferrofluids is much higher than that of ordinary solid paramagnets, so that it promptly reacts to even minor magnetic stimuli. This behavior leads to the development of a number of interesting field-induced interfacial instabilities, and pattern formation processes which have attracted much interest [44, 45, 46, 47, 48].

One striking example of pattern-forming systems in ferrofluids is the popular Rosensweig (or, peak) instability [49]. It occurs when a uniform magnetic field is applied normal to an initially flat, ferrofluid free surface. The competition between magnetic, gravitational, and cap-illary forces results in the rising of a three-dimensional (3D) array of spiky structures, that look like horns growing from the liquid free surface. During the last four decades both linear and nonlinear aspects of the problem have been vigorously investigated [49, 50, 51, 52, 53, 54]. Recent variations of this archetypal ferrohydrodynamic instability revealed other exceptional properties such as the formation of stable solitonlike structures at the magnetic fluid-air inter-face [55], the verification of a hybrid-type instability in miscible ferrofluids where peak and labyrinthine patterns arise [56], and the occurrence of magnetic wave turbulence on the surface of the ferrofluid [57].

An effectively 2D counterpart of the traditional 3D Rosensweig instability can be obtained when a more viscous and more dense ferrofluid is placed below a nonmagnetic fluid in the con-fined geometry of a vertical Hele-Shaw cell. Experimental studies [58, 59] have demonstrated that the initially flat fluid-fluid interface goes unstable if a uniform magnetic field is applied normal to it, and in the plane of the Hele-Shaw cell. It has been shown that above a certain

critical value of the applied field, the interface deforms leading to the formation of a regular pattern formed by a periodic line of quasi-2D peaked structures. In Ref. [59] a Darcy’s law approach is used to describe the early dynamics of such ferrofluid peak arrangement. The ini-tial evolution of small interfacial deformations has been studied by a linear stability analysis, in which the condition for the neutrality of such deformations was determined. However, not much has been discussed about the development of such confined peak-shaped morphology at nonlinear stages of the dynamics.

A related theoretical investigation [52] tried to mimic the fully 3D Rosensweig problem by focusing on an idealized version of the system. A high-order perturbative approach has been employed to study the static surface profile of a vertical, truly 2D ferrofluid layer subjected to a normal magnetic field. The starting point of their analysis assumes that the shape of the perturbed ferrofluid interface is determined by a pressure equilibrium condition. In this context, a somewhat cumbersome Galerkin-type anzatz is used to expand both the magnetic field and the surface deflection up to fifth-order in the perturbation amplitudes. As a result, peaks are obtained for larger values of the applied magnetic field. Despite the relative intricacy of their analytical method, no specific mechanism is proposed to explain the formation of the static peaks within the nonlinear regime. Moreover, the nonlinear stability of such static structures has not been analyzed.

2.1

Normal-field instability in confined ferrofluids

In this section we show that the phenomenon of ferrofluid peak formation in vertical Hele-Shaw cells under a normal, in-plane magnetic field can be properly reproduced at lowest non-linear perturbative order through a mode coupling approach of the dynamics [60, 61]. By employing a second-order theory and considering the interplay of a small number of Fourier modes, we show that the main features of the ferrofluid peak formation in confined geome-try can be revealed in a very simple and clear manner. The nonlinear coupling is due to the influence of a normal magnetic traction term which appears in a generalized Young-Laplace

Figure 2.1: Schematic illustration of the vertical Hele-Shaw cell setup. Fluid 1 is a ferrofluid (shaded region), while fluid 2 is nonmagnetic. The densities and viscosities of the fluids are respectively denoted byρj, andηj, where j= 1 and 2. A uniform magnetic field H0is applied

along the positive y direction, and the acceleration of gravity points downward (g_{= −gˆy). The} cell has thickness b, and interfacial perturbations are represented byζ =ζ(x,t).

pressure drop boundary condition for ferrofluids. We have also studied the stationary shapes obtained when the forces involved balance equally. This offers the opportunity to contrast the shapes of time evolving and steady state structures. Nonlinear stationary solutions are found to be stable.

2.1.1 Mode coupling strategy

Consider a vertical Hele-Shaw cell of thickness b containing two semi-infinite immiscible viscous fluids. Denote the densities and viscosities of the lower and upper fluids, respectively as ρ1, η1 and ρ2, η2 (Fig. 2.1). The cell lies parallel to the xy plane, where the y axis is

vertically upward. Between the two fluids there exists a surface tensionσ, and the lower fluid is assumed to be a ferrofluid (magnetization M), while the upper fluid is nonmagnetic (zero magnetization). Acceleration of gravity g_{= −gˆy, where ˆy denotes the unit vector in the y-axis.}

A uniform external magnetic field H0= H0ˆy is applied in the plane of the cell, being normal to

the initially flat fluid-fluid interface.

Due to the action of the magnetic field the fluid-fluid interface may deform, and its per-turbed shape is described as I_{(x, y,t) = y −}ζ(x,t) = 0, whereζ(x,t) =∑+_k=−∞ _∞ζk(t) exp(ikx)

represents the net interface perturbation with Fourier amplitudesζ_k(t), and wave numbers k. We follow the standard approximations used by other investigators [14, 15, 16, 17, 62, 63] and assume that the ferrofluid is magnetized such that its magnetization is constant and collinear with the applied field M(H) = M(H0)ˆy. We consider only the lowest order effect of the

mag-netic interactions that would result in fluid motion.

For the confined geometry of a Hele-Shaw cell, we reduce the 3D flow to an equivalent 2D one by averaging the Navier-Stokes equation over the direction perpendicular to the plates (defined by the z axis). Using no-slip boundary conditions and neglecting inertial terms, one derives a modified Darcy’s law as [17, 38, 59, 62]

vj= − b2 12ηj  ∇pj− 1 b Z +b/2 −b/2 µ0(M ·∇)Hdz +ρjgˆy  (2.1)

where j= 1 ( j = 2) labels the lower (upper) fluid, and pjdenotes the hydrodynamic pressure.

The local magnetic field appearing in (2.1) differs from the applied field H0by a demagnetizing

field of the polarized ferrofluid H= H0+ Hd, where Hd = −∇ϕ, andϕ is a scalar magnetic

potential. Notice that since the applied field is spatially uniform it eventually drops out in the calculation of the magnetic term in (2.1), and the magnetic effects are due to the demagnetizing field.

As commented earlier we consider that the magnetization of the magnetic fluid in a uniform magnetic field is both uniform and constant, an assumption first introduced by Cebers and Maiorov [16]. This corresponds to method C in Ref. [64], where the validity of the constant magnetization hypothesis has been examined. We emphasize that although the magnetization is assumed to be uniform, the demagnetizing field is not. It is precisely this shape dependent demagnetizing field contribution (the so-called “fringing field") that gives rise to the fingering

instabilities in this model (see, for instance, Refs. [17, 63]). The influence of the constant magnetization approximation on the nature of the flat-deformed interface transition is discussed in Ref. [65] for the ferrofluid labyrinthine pattern formation case.

Equation (2.1) can be conveniently rewritten as

vj= − b2 12ηj ∇pj+µ0 M b Z +b/2 −b/2 ∂ϕ ∂ydz+ρjgy  (2.2) where ϕ = 1 4π Z S M_{· n}′ |r − r′_|d 2_r′₌ 1 4π Z +∞ −∞ Z +b/2 −b/2 M ˆy_{· n}′dx′dz′ p(x − x′₎2_{+ (y − y}′₎2_{+ (z − z}′₎2. (2.3)

The unprimed coordinates r denote arbitrary points in space, and the primed coordinates r′are integration variables within the magnetic domain S , and d2r′= dx′dz′denotes the infinitesimal area element. The vector n′represents the unit normal to the magnetic domain in consideration. In Eq. (2.2) the velocity depends on a linear combination involving gradients of hydrodynamic pressure, magnetic potential, and gravity term, so we may think of the term between curly brackets as an effective pressure.

From Eq. (2.2) and the incompressibility condition ∇_{· v}j = 0 it can be verified that the

velocity potentialφj(vj= −∇φj) obeys Laplace’s equation. The problem is then specified by

the augmented pressure jump boundary condition at the interface

p1− p2=σκ−

1

2µ0(M · n)

2

, (2.4)

plus the kinematic boundary condition, which states that the normal components of each fluid’s velocity are continuous at the interface

The first term on the right-hand side of Eq. (2.4) represents the usual contribution related to surface tension and interfacial curvatureκ. The second term is the so-called magnetic normal traction [14, 15], which considers the influence of the normal component of the magnetization at the interface. For the current field configuration this magnetic piece is at least of second-order in the interface perturbationζ, being legitimately nonlinear. The magnetic traction term will have a key role in determining the shape of the emerging interfacial patterns at the onset of nonlinear effects.

We proceed by following standard steps performed in weakly nonlinear studies [60, 61]. First, Fourier expansions are defined for the velocity potentials, and then boundary condi-tions (2.4) and (2.5) are used to express φj in terms of ζk consistently up to second-order.

By substituting these relations in Eq. (2.2), and Fourier transforming, yields a dimensionless mode coupling equation for the system (for k_{6= 0)}

˙ ζk=λ(k)ζk+

∑

NB= µ0

M2b

σ (2.8)

represents a magnetic Bond number, and measures the ratio of magnetic to capillary forces. Likewise,

NG=

(ρ1−ρ2)gb2

σ (2.9)

rela-tive to the surface tension. In addition, W(k) = 1 π Z ∞ 0  sinτ τ 2 [ q (k/2)2₊τ2₋τ_{] d}τ _(2.10)

is clearly a positive quantity, and originates from the contribution of the demagnetizing field. The second-order mode coupling terms are given by

F(k, k′) =NB 2 |k|k

′_{(k − k}′_{), (2.11)}

G(k, k′_{) = A|k|[sgn(kk}′_{) − 1].} (2.12)

The sgn function equals_{±1 according to the sign of its argument, and the viscosity contrast} is defined as A= (η1−η2)/(η1+η2). In Eqs. (2.6)-(2.12) lengths and velocities are rescaled

by b, and σ/[12(η1+η2)], respectively. We focus on the situation in which ρ1 >ρ2, and

η1≫η2, so that the fluid-fluid interface is gravitationally stable (NG> 0) and A ≈ 1. This is

done to allow a more direct connection with existing experiments [58, 59] whereη1≫η2. As

a matter of fact, the second-order results presented in the rest of this section remain practically unchanged as A is modified.

It is worth noting that the coupling term (2.11) comes from magnetic normal traction con-tribution in the pressure jump condition (2.4). This is exactly the term that is responsible for the development of peaked ferrofluid patterns already at second-order. Observe that there is no demagnetizing field contribution at second-order. We stress that the theoretical results pre-sented in the following sections utilize dimensionless quantities which are extracted from the realistic physical parameters used in the experiments of Ref. [59].

2.1.2 Pattern morphology and nonlinear stability of stationary patterns

Before examining how we can use the mode coupling equation (2.6) to access purely non-linear aspects related to the morphology of the interface, we briefly discuss a few useful con-cepts associated with the linear growth rate (2.7). Since a positiveλ(k) leads to an unstable

0 2 4 k -3 0 3 6 ΛH kL kc k* kt NB=13.83 NB=11.12 NB=9.50

Figure 2.2: Linear growth rate λ(k) as a function of the wave number k for NG= 1.44, and

three different values of the magnetic Bond number NB. The critical (kc), fastest growing (k∗),

and threshold (kt) wave numbers are also indicated. The critical magnetic Bond number is

NB= 11.12.

behavior, Eq. (2.7) tells us that the magnetic terms NB and W(k) are destabilizing. On the

other hand, gravity and surface tension try to stabilize interfacial disturbances. The interplay of these competing effects determines the linear stability of the flat interface. This is illustrated in Fig. 2.2 which plotsλ(k) in terms of k, for NG= 1.44, and three increasingly larger values

of the magnetic Bond number. It is clear from Fig. 2.2 that the transition from a stable to an unstable situation occurs if both of the following conditions are met

λ(k) = 0, and ∂λ(k) ∂k = 0.

This defines a critical wave number kc and a critical magnetic Bond number at which this

exchange of stability takes place. Thus, when NB is increased from zero, the interface remains

flat over a range of values of NB up to a point when transition suddenly occurs (NB= 11.12),

59]. Note that the experimental investigation performed in Ref. [58] reveals the emergence of nonzero height peaks at the critical field, and a hysteresis phenomenon is observed. However, this effect is not contemplated by our analytical model, and also not detected in Ref. [59].

For larger NB two other quantities of interest can be defined for a range of wave numbers

over whichλ_{(k) ≥ 0 (see Fig. 2.2): k}∗, the wave number of the fastest growing mode, max-imizesλ(k); kt, the threshold wave number beyond which all modes are stable, is the largest

wave number for whichλ(k) vanishes. As NB grows, k∗and kt shift to the right and modes of

higher wave number become unstable.

At this point, we turn our attention to the weakly nonlinear, intermediate stages of pattern evolution, and use the equation of motion (2.6) to investigate how the magnetic field influences the shape of the fingering patterns at the onset of nonlinear effects. Inspired by an approach originally proposed in Refs. [60, 61], we focus on a mechanism controlling the finger shape behavior through magnetic means, and consider the coupling of a small number of modes. For a given NBlarger than the critical value, only discrete modes multiple of k∗are selected. In this

framework, we examine the interaction of the fundamental mode with its own harmonic. For the rest of this section, we consider NB above the critical situation, and take the fundamental

wave number k= k∗as the fastest growing mode. Consequently, the harmonic mode 2k∗always lies to the right of the threshold wave number kt. Therefore, the harmonic is always linearly

stable against growth.

As extensively discussed in Refs. [24, 60, 61] when one considers the second-order coupling of just two modes (i.e., the fundamental and its harmonic) one finds that the presence of the fundamental naturally forces growth of the harmonic mode through a nonlinear driven term in the mode coupling equations. The interesting point is that the sign of such nonlinear driven term dictates whether fingertip sharpening or fingertip broadening is favored by the dynamics. If the nonlinear term is positive, the harmonic mode is driven positive, the sign that is required to cause upward pointing fingers to become sharp, favoring fingertip sharpening. In contrast, if the nonlinear coupling term is negative, growth of a negative harmonic is favored, leading to upward-pointing fingertip broadening. Based on this mechanism the finger shape behavior of confined fluid systems can be described in a very simple manner.

It turns out that the ferrofluid finger tip-sharpening behavior observed experimentally in Refs. [58, 59] can be described by the mechanism mentioned above. The justification is that finger tip-sharpening requires the growth of a sizable harmonic mode. Of course, this cannot be achieved through a purely linear description. However, by inspecting Eq. (2.6), we note that the second-order term F(n, n′), which is reminiscent of the magnetic normal traction contribution in (2.4), does involve magnetic field effects. This term drives growth of the harmonic mode (with the phase appropriate to sharpen finger tips) despite its linear stability, leading to the development of peaked ferrofluid structures already at second-order.

Considering such a mechanism our aim is to illustrate the time evolution of the interface and the occurrence of ferrofluid finger tip-sharpening. It is convenient for the subsequent dis-cussions to consider cosine ak =ζk+ζ_−k and sine bk= i(ζk−ζ_−k) modes, rather than the

complex modes employed in Eq. (2.6). Describing the fundamental as a cosine mode with positive amplitude, we only need to examine the harmonic cosine mode to analyze finger tip behavior and pattern morphology. Under such circumstances, the evolving interface can be described asζ(x,t) = ak(t) cos kx + a2k(t) cos2kx, where the perturbation amplitudes ak(t) and

a2k(t) are obtained from the second-order solution of the mode coupling Eq. (2.6). Specifically,

one needs to solve the following coupled nonlinear differential equations

˙ a2k = λ(2k)a2k+ 1 2[F(2k, k)ak+ G(2k, k) ˙ak] ak, (2.13) ˙ ak=λ(k)ak+ 1 2[F(k, −k)ak+ G(k, −k) ˙ak] a2k+ 1 2[F(k, 2k)a2k+ G(k, 2k) ˙a2k] ak. (2.14)

By solving Eqs. (2.13) and (2.14) the interface evolution is plotted in Fig. 2.3 for NG= 1.44,

NB= 13.83, k∗= 3, and considering initial conditions ak(0) = 0.0001, and a2k(0) = 0. Note

that the harmonic mode is absent initially. The times shown are t= 0.8, 1.2, and 2. The re-sulting patterns reveal the emergence of increasingly sharp, peaked structures of the ferrofluid

Figure 2.3: Time evolution of the interface shape. The shaded region represents the ferrofluid pattern morphology at time t= 2. The interface profile for t > 2 is indistinguishable from the one shown at t= 2.

penetrating the nonmagnetic fluid. As time increases the downward moving fingers of the up-per fluid look wider and flatter at their extremities, getting closer to the stationary shape already revealed at t= 2 (shaded region in Fig. 2.3). This characteristic shape is consistent with the ex-perimental patterns exhibited in Refs. [58, 59], and also with the purely static profiles obtained theoretically in Ref. [52]. It is also worthwhile to note that the interface profiles obtained for

t> 2 lie on top of the curve plotted at time t = 2 signalizing that a steady state has been reached. Considering the simplicity of our analytical approach (lowest-order nonlinear coupling of just two Fourier modes), as expected we find that a precise quantitative agreement between our the-oretical shapes and the experimental profiles is not observed. However, the main morphological features of the real ferrofluid patterns can be indeed predicted and satisfactorily depicted by our mode coupling theory.

Complementary information about the pattern-forming phenomenon depicted in Fig. 2.3 is provided by Fig. 2.4 which plots the time evolution of the cosine perturbation amplitudes ak

and a2k. We clearly observe that the weakly nonlinear coupling dictates naturally the enhanced

growth of a positive harmonic mode, precisely the phase that is required to produce finger tip-sharpening. It is also evident that after an initial period of growth both perturbation amplitudes

saturate, so that they remain unchanged as time progresses. This confirms the idea that the system tends to a steady state configuration.

0 1 2 3 t 0 0.04 0.08 Perturbation amplitudes ak a2 k

Figure 2.4: Time evolution of the perturbation amplitudes ak(t) and a2k(t) for the evolving

interface depicted in Fig. 2.3. It is clear that both amplitudes eventually tend to stationary values.

The nontrivial stationary cosine amplitudes for the harmonic (ast_2k) and fundamental (ast_k) can be obtained analytically by setting their time derivative terms to zero in the mode coupling equations (2.13) and (2.14), yielding

ast_2k= −2λ(k) [F(k, −k) + F(k,2k)], (2.15) ast_k = s 4λ(2k)λ(k) F_{(2k, k)[F(k, −k) + F(k,2k)]}, (2.16) where F_{(k, −k) + F(k,2k) = −2N}Bk2|k| < 0,

F(2k, k) = NBk2|k| > 0,

λ(k) > 0, andλ(2k) < 0.

It is reassuring to observe that ast_2kis a genuinely positive quantity.

We close this section by examining the stability of the stationary solutions. In fact, we have verified that the steady solution given by Eqs. (2.15) and (2.16) is stable. This is done by con-sidering the nonlinear differential equations (2.13) and (2.14). Through a standard linearization process close to the stationary solution, we diagonalize the resulting system of equations, de-termining the eigenvalues which dictate the stability of the fixed point [66]

ε±= λ (2k) 2 ( 1_± s 1+ 16k 2_N Bλ(k) λ(2k)[λ(k) + 2k2_N B] ) . (2.17)

For the stationary solution under consideration both eigenvalues have negative real parts, char-acterizing a stable node or spiral. Regardless of the initial conditions for the perturbation am-plitudes the system asymptotically approaches the stable fixed point. It is worth pointing out that this is in contrast with the typical unstable behavior exhibited by other steady (nonzero surface tension) solutions obtained for nonmagnetic [67, 68] and magnetic [24, 40, 41] fluids in Hele-Shaw cells. We also find that the unperturbed (flat) interface is a saddle point, and therefore unstable.

2.2

Nonlinear traveling waves in confined ferrofluids

Frontal fluid flows in the confined geometry of a Hele-Shaw cell have been plentifully in-vestigated during the last five decades. Under frontal flow, the motion of the fluids is normal to the initially undisturbed interface between them, and might lead to the formation of viscous fingering phenomena [1, 2]. Curiously, the related problem of fluids flowing parallel to their

separating interface, and the development of interfacial traveling waves in the Hele-Shaw setup has been much less exploited in the literature [69, 70, 71, 72]. Zeybek and Yortsos [69, 70] studied parallel flow in a horizontal Hele-Shaw cell. In the limit of large capillary numbers and large wavelength they have found Korteweg-de Vries (KdV) dynamics leading to stable finite amplitude soliton solutions. Afterward, Gondret, Rabaud, and co-workers [71, 72] exam-ined, through experiments and theory, the appearance of traveling waves for parallel flow in a vertical Hele-Shaw cell. They have observed that the interface is destabilized above a certain critical flow velocity, so that waves grow and propagate along the cell. Such waves are initially sinusoidal then turn to localized structures presenting a nonlinear shape.

The theoretical model presented in Ref. [71] was based on a modified Darcy equation for the gap-averaged flow with an additional term representing inertial effects. Within this context a Kelvin-Helmholtz instability for inviscid fluids has been found. For viscous fluids they derived a Kelvin-Helmholtz-Darcy equation and verified that the threshold for instability was governed by inertial effects, while the wave velocity was determined by the Darcy’s law flow of viscous fluids. Their theoretical analysis has been backed up by their own experimental results. Theo-retical improvements in the description of the system have been proposed in Refs. [73, 74, 75] where the gap-averaged approach utilized in [71] has been replaced by an alternative scheme directly based on the fully three-dimensional Navier-Stokes equation. In the end, the calcula-tions in Refs. [73, 74, 75] lead to an equation of motion similar to the one derived in [71], but with slightly different coefficients.

One additional example of parallel flow in vertical Hele-Shaw cells is the linear stability analysis performed by Miranda and Widom [76]. The major difference between their work and the ones performed in Refs. [69, 70, 71, 72, 73, 74, 75] is the fact that one of the fluids is a ferrofluid [14, 15], and that an external magnetic field is applied. The field could lie in the plane of the Hele-Shaw cell, either tangential or normal to the fluid-fluid interface. A ferrofluid behaves as a regular viscous fluid except that it can experience forces due to magnetic polarization [77]. This opens up the possibility of investigating the role played by the magnetic field in the dynamics of the parallel flow. It has been shown [76] that the dispersion relation governing mode growth is modified so that the magnetic field can destabilize the interface even

in the absence of inertial effects. However, it has been deduced that the magnetic field would not affect the speed of wave propagation. Despite all that, a study addressing the effect of the magnetic field on the morphological structure and nonlinear evolution of the propagating waves is still lacking.

In this section we re-examine the problem initially proposed in Ref. [76] by considering the action of an in-plane, tilted applied magnetic field which makes an arbitrary angle with the direction defined by the unperturbed fluid-fluid interface. This apparently naive modification proves to be crucial in creating a connection between the applied field and the propagating wave velocity. Moreover, in contrast to what was done in [76] we go beyond the linear regime, and tackle the problem by using a perturbative weakly nonlinear approach. This particular theoreti-cal tool enables one to extract valuable analytitheoreti-cal information at the onset of nonlinearity. As a consequence, one can investigate the influence of the magnetic field on the nonlinear dynamics and ultimate shape of the traveling surface waves.

The layout of this section is as follows. Section 2.2.1 introduces the governing equations of the parallel flow system with a ferrofluid, and presents our mode-coupling approach which is valid at lowest nonlinear perturbative order [60, 61]. Linear and weakly nonlinear dynamics are discussed in Secs. 2.2.2 and 2.2.3. We show that the effect of the magnetic field on the velocity and shape of the propagating waves can be accessed by considering the interplay of a small number of Fourier modes. One important result is the feasibility of sustaining, moving, and controlling a traveling wave solely under the action of an external magnetic field. Station-ary wave profiles are found for different values of the magnetic field tilting angle. Our main conclusions are summarized in Sec. 6.1.

2.2.1 Governing equations and analytical calculations

Consider two semi-infinite immiscible viscous fluids, flowing with velocities Ujwhere j=

1 ( j= 2) labels the lower (upper) fluid. The flow takes place along the x direction in a vertical Hele-Shaw cell of thickness b (Fig. 2.5). The densities and viscosities of the fluids are denoted respectively asρj, andηj. The cell lies parallel to the xy plane, where the y axis is vertically

upward. Between the fluids there exists a surface tensionσ, and the lower fluid is assumed to be a ferrofluid (magnetization M), while the upper fluid is nonmagnetic (zero magnetization). Acceleration of gravity g_{= −gˆy, where ˆy is the unit vector in the y-axis. The base flow is} horizontal with η1U1=η2U2 [71] because the flows in the two fluids are driven by the same

pressure gradient.

A uniform external magnetic field H0= H0(cosα ˆx+ sinα ˆy) is applied in the plane of the

cell. The shape of the perturbed fluid-fluid interface is described as I_{(x, y,t) = y −}ζ(x,t) = 0, whereζ(x,t) =∑+_k=−∞ _∞ζk(t) exp(ikx) represents the net interface perturbation with Fourier

amplitudesζk(t), and wave numbers k.

For the quasi two-dimensional (2D) geometry of the Hele-Shaw cell, the 3D fluid flow is reduced to an equivalent 2D one by averaging the Navier-Stokes equation over the direction perpendicular to the plates. Using no-slip boundary conditions and neglecting inertial terms, the flow in such a confined environment is governed by the modified Darcy’s law in Eq. (2.1).

Figure 2.5: Schematic configuration of the parallel flow in a vertical Hele-Shaw cell. The lower fluid is a ferrofluid, while the upper fluid is nonmagnetic. An external uniform magnetic field

H0is applied making an angleα with the initially undisturbed interface separating the fluids.

The role of inertia in the problem can be quantified by a Reynolds number (relative measure of inertial and viscous forces) which is directly proportional to the cell gap thickness, and

in-versely proportional to the viscosity of fluid, Rej= (ρjUjb)/(12ηj). Since most experimental

and theoretical studies of ferrofluid flow in Hele-Shaw cells deal with very thin cell gaps and highly viscous fluids, the vanishing Reynolds number limit is readily validated. Under such circumstances, the fluid motion is perfectly described by the gap-averaged modified Darcy’s law (2.1). As discussed in Refs. [71, 78, 79] in unidirectional Hele-Shaw parallel flow, the in-ertial effects can be neglected, even at relatively large Reynolds numbers as long as Rej< Rec,

where Recis the Reynolds number corresponding to the laminar-turbulent transition.

We follow the standard approximations used in the previous section and assume that the magnetization is collinear with the applied field M(H) = M(cosα ˆx+ sinα ˆy), where M = M(H0). Only the lowest order effect of the magnetic interactions that would result in fluid

motion is considered. We emphasize that although the magnetization is taken to be uniform, the demagnetizing field is not.

Taking into consideration the physical assumptions mentioned above, and the particular geometry of our system Eq. (2.1) can be rewritten as

vj= − b2 12ηj ∇ ( pj+ρjgy+µ0 M b Z +b/2 −b/2  cosα∂ϕ ∂x + sinα ∂ϕ ∂y  dz ) , (2.18) where ϕ = 1 4π Z S M_{· n}′ |r − r′_|d 2 r′ = 1 4π Z +∞ −∞ Z +b/2 −b/2

M(cosα ˆx+ sinα ˆy_{) · n}′dx′dz′ p(x − x′₎2_{+ (y − y}′₎2_{+ (z − z}′₎2.

(2.19)

The unprimed coordinates r denote arbitrary points in space, and the primed coordinates r′are integration variables within the magnetic domain S , and d2r′= dx′dz′denotes the infinitesimal area element. The vector n′represents the unit normal to the magnetic domain under study.

By inspecting Eq. (2.18) we observe that the velocity depends on a linear combination involving gradients of hydrodynamic pressure, gravity, and magnetic potential, respectively.

So, the term between curly brackets in (2.18) can be seen as an effective pressure. Therefore, as in the Hele-Shaw problem with nonmagnetic fluids [1, 2], the flow is potential, vj= −∇φj,

but now with a velocity potential given by

φj= b2 12ηj  pj−µ0M2I(x, y) +ρjgy , (2.20) where I(x, y) = 1 4πb Z +∞ −∞ Z +b/2 −b/2 Z +b/2 −b/2 h −cosα ∂ζ_∂(xx′′)+ sinα i r 1+∂ζ_∂(x_x′′) 2 ×[cosα (x − x ′_{) + sinα (y −ζ(x}′_{))] dx}′_dz′_dz p(x − x′₎2_{+ (y −}ζ_(x′₎₎2_{+ (z − z}′₎2 . (2.21)

In Eq. (2.21) the integral in dz is related to the gap-average calculation [see Eq. (2.18)], while the integrals in dx′ and dz′ come from the surface integral in the magnetic domain of interest S [see Eq. (2.19)]. Notice that incompressibility (∇_{· v}j= 0) then yields Laplace’s equation

for the velocity potential.

The problem is specified by the two boundary conditions given by Eqs. (2.4) and (2.5). Equation (2.4) is an augmented pressure jump condition at the interface, whereκ denotes the interfacial curvature. A crucial difference of this expression from the one utilized in the non-magnetic situation is given by the second term on the right-hand side: the so-called non-magnetic normal traction [14, 15] which considers the influence of the normal component of the mag-netization at the interface. For the current field configuration this magnetic piece is at least of second-order inζ, being legitimately nonlinear. This magnetic term contributes to determine the shape of the traveling wave profiles at the onset of nonlinear effects. The second boundary condition (2.5) simply states the continuity of the normal flow velocity at the interface.

Our next task is to derive an equation of motion for the perturbation amplitudes which is able to capture the essential physics at the lowest nonlinear level. This is done by following standard steps performed in previous weakly nonlinear studies [38, 60, 61]. First, we define

Fourier expansions for the velocity potentials. Then, we expressφj in terms of the

perturba-tion amplitudesζk by considering the kinematic boundary condition (2.5). Substituting these

relations, and the modified pressure jump condition Eq. (2.4) into Eq. (2.20), always keeping terms up to second-order inζ, and Fourier transforming, we find the dimensionless equation of motion (for k_{6= 0)} ˙ ζk = Λ(k)ζk +

∑

where the overdot denotes total time derivative,

Λ(k) =λ_{(k) − ik}  c0+ NB|k| sin 2α 2  (2.23)

is a complex linear growth rate, and

λ(k) = |k|{NB[sin2α W1(k) − cos2αW2(k)]

− k2− NG} (2.24)

is its real part.

The system is characterized by three dimensionless parameters

NB = µ0 M2b σ , NG= (ρ1−ρ2)gb2 σ , and c0 = 12(η1U1+η2U2) σ .

The magnetic Bond number NB measures the ratio of magnetic to capillary forces, while the

gravitational Bond number NGaccounts for the relative importance of gravitational force to the

surface tension. The parameter c0 represents the propagation contribution due to the parallel

12ηjUj/σ is the capillary number of fluid j. In addition, W1(k) = 1 π Z ∞ 0 (1 − coskτ) τ2 [ p τ2_{+ 1 −}τ_{] d}τ_{, (2.25)} and W2(k) = k π Z ∞ 0 sin kτ τ [ p τ2_{+ 1 −}τ_{] d}τ _(2.26)

originate from the contribution of the demagnetizing field. The second-order mode-coupling terms are given by

F(k, k′) = NB|k| k′(k′_{− k)} 2 ( cos 2α + i sin 2α W1(k′) k′ − W2(k) 2k + W3(k, k′) k′(k′_{− k)} ) , (2.27) and G(k, k′_{) = A|k|[sgn(kk}′_{) − 1],} (2.28) where W3(k, k′) = 1 π Z ∞ 0

[sin kτ_{− sink}′τ_{− sin(k − k}′)τ]

× ( [√τ2_{+ 1 −}τ_] τ4 + 1 2τ  1 τ − 1 √ τ2_{+ 1} ) dτ (2.29)

is another demagnetizing integral. Notice the presence of the imaginary part in (2.27), that is proportional to NB and sin 2α, and would vanish for a purely vertical or horizontal magnetic

field. The sgn function equals _{±1 according to the sign of its argument, and the viscosity} contrast is defined as A= (η1−η2)/(η1+η2). In Eqs. (2.22)-(2.29) lengths and velocities

are rescaled by b, and σ/[12(η1+η2)], respectively. Without loss of generality we focus on

(NG> 0) and A ≈ 1. Since the gravitational Bond number plays a minor role to our analysis,

for the rest of this section we fix its value as NG= 1.4. We recover the results for the vertical

magnetic field configuration without flow, previously studied in Ref. [38], by settingα =π/2 and c0= 0. It should be noted that the theoretical results presented in the following section

utilize dimensionless quantities which are extracted from the realistic physical parameters used in the experiments of Refs. [71] and [59].

0 5 10 15 k -50 0 50 100 150 200 250 ΛH kL NB=30 Α=1.06 Α=1.31 Α=А2 NB=20 Α=1.17 Α=1.30 Α=А2

Figure 2.6: The real part of the linear growth rateλ(k) as a function of the wave number k for NG= 1.4. Continuous (dashed) curves refer to NB= 20 (NB= 30). For each NBwe plot curves

for three values of the angleα, where lighter gray curves correspond to higher values ofα.

2.2.2 Linear regime

Before examining how we can use the mode-coupling equation (2.22) to access important nonlinear aspects related to the traveling waves, we briefly discuss a few useful concepts as-sociated with the linear growth rate (2.23). The real part of the growth rate Re[Λ(k)] =λ(k) governs the exponential growth or decay of the wave amplitudes at the linear regime. Since a positiveλ(k) leads to an unstable behavior, Eq. (2.24) tells us that the term W1(k) [W2(k)],